Procedure for Delocalization of Test Data and Material Identification
Until very recently it has been general practice to identify the postpeak stress-strain relation from test data ignoring the fact that the deformation of the specimen within the gage length often becomes nonuniform, due to localization of cracking damage. The fact that damage must localiz. e, except in the smallest possible specimens, was shown in detail in Chapter 8. The correct analysis of localization in strain softening materials led first to the development of the crack band model (ChapterS), and later to the more sophisticated models described in the preceding chapter. The localization phenomena were already documented in the early eighties. However, because the general problem of identification of material parameters in presence of strain-softening localization (Ortiz 1987) is tremendously complex, the contamination of test
data by localization has typically been ignored. At the present state of knowledge, however, this is no longer acceptable. The data must be decontaminated, delocalized. An approximate procedure to do that, applicable to any type of constitutive model, was recently proposed by Bazant, Xiang et al. (1996).
The delocalization cannot, and need not, be done with a high degree of accuracy and sophistication. In the identification of the microplane model by Bazant, Xiang et al. (1996), the test data from laboratory specimens have been analyzed taking into account the strain localization in an approximate manner. The idea is to exploit two simple approximate concepts: (1) localization in the series coupling model described in Sections 8.1-8.3, and (2) the effect that energy release due to localization within the cross section of specimen has on the maximum load, as described by Bazant’s size effect law (Section 1.4 and Chapter 6).
The strain as commonly observed is the average strain em on a gage length L. According to the series coupling model and the crack band model (Sections 8.1-8.3), the strain may be assumed to localize after the peak into a band of width hc, as depicted in Fig. 14.2.1a, while the remainder of the gauge length unloads. In this way, the strain of the material inside the localized zone is es – – corresponding to the softening branch— while in the remaining part the strain is Eu, as given by the unloading curve from the peak (Fig. 14.2.1b).
The strain that the constitutive model for damage should predict is the strain es in the localization zone. But this strain is difficult to measure, for three reasons: (1) the size of the localization zone is small, which reduces the accuracy of strain measurements; (2) the location of the localization zone is uncertain, and so one does not know where to place the gage; and (3) the deformation of the localization zone is quite random while the constitutive model predicts the statistical mean of many random realizations (determining this mean requires taking measurements on many specimens). Therefore, a simplified method is desirable based on measuring only the average strain £m.
To find the simplified formula, we note that the total increment of the gauge length AL (equal to Lem by definition) is obtained by adding the contributions of the softening and unloading regions, i. e.,
L/Є in — hcEs + (L – /tc)cu
If we further assume that the unloading proceeds parallel to the initial elastic loading (i. e., stiffness degradation up to the peak is negligible), then the unloading strain is єи =■• ev — (су, – cr)/E, where E is the elastic modulus and Ev and cp are the strain and stress at the peak of the stress-strain curve for the given type of loading (Fig. 14.2.1b). So we finally get
To correct the given test data according to (14.2.2), one must obviously know the value of the localization length hc. It is impossible to determine this length from the reports on the uniaxial, biaxial, and triaxial tests of concrete found in the literature. However, a reasonable estimate can be made by experience front other studies; l ~ 3da where da — maximum size of the aggregate in concrete (for high-strength concretes, і is likely smaller, perhaps as small as 1 = da).
Bazant, Xiang et ai. 1996 further proposed an approximate procedure to filter out of the given tensile test data the size effect on the maximum tensile stress. This procedure was based on the size effect law. According to the size effect on maximum load, they scaled the measured response curve by affinity transformation with respect to the strain axis and in the direction parallel to the elastic slope. Thus, they obtained the response curve with the peak tensile stress corresponding to specimen of size hc.